Volume Fluctuations and Geometrical Constraints in Granular Packs

I study the structural organization and correlations in very large packings of equally sized spheres, reconstructed in three dimensions with x-ray computed tomography. I show that the geometrical structure can be conveniently studied as a packing of irregular tetrahedra with volume distribution that must decay exponentially with parameters controlled by the conditions of mechanical stability, nonoverlapping, and space filling. I argue that the system’s structure can be described as constituted of two phases: (1) an “unconstrained” phase which freely shares the volume and (2) a “constrained” phase which assumes configurations accordingly with the geometrical constraints. It results that the granular system exploits heterogeneity maximizing freedom and entropy while constraining mechanical stability.

Figure 1

Dihedral angle distribution ($x$ axis: angular degrees; $y$ axis: renormalized frequencies). The vertical lines indicate the angles $\theta =n\arccos (1/3)$ (and $360-\theta$) with $n=1$, 2, 3, 4, 5 (tetrahedral packings). The dashed lines are at the angles $\theta =n360/5$ ($n=1$, 2, 3, 4).

Figure 2

(a) Normalized frequencies of the distribution of Delaunay volumes in G. The vertical lines indicate the maximum volumes attainable by tetrahedra with 4, 5, or 6 couples of spheres in contact (respectively: $\sqrt{3}{d}^3/12$, $d^3/8$, and $\sqrt{2}{d}^3/12$). (b) Log-log plots of the cumulative distributions $P_{<}(v)$ (the probability of finding a volume smaller than $v$). (c) Log-linear plots of the inverse normalized cumulative distribution $P_{>}(v)$ (the probability of finding a volume larger than $v$).

Figure 3

Coefficients ${\beta }^{-1}/d^3$ vs the inverse packing fraction ${\rho }^{-1}$. The symbols correspond to the six samples A–F. The full line is the theoretical predictions from Eq. (4) with $⟨f⟩=14.5$. The two dot-dashed lines are the theoretical results for the two extreme cases $⟨f⟩=14.3$ and $⟨f⟩=14.6$.